Lesson 11

The purpose of this warm-up is to elicit the idea that not every set of three pairs of congruent corresponding parts will guarantee triangle congruence, which will be useful when students explore Side-Side-Angle Triangle Congruence in a later activity. While students may notice and wonder many things about these statements and images, the fact that the triangles are not congruent despite having so many corresponding parts congruent is the important discussion point. 

First, display the congruence statements without the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner.

Next, display the image for all to see. Give students 1 minute of quiet think time, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students’ surprise at the images does not come up during the conversation, ask students to discuss this idea.

In this activity, students are arranged in groups of three, given side lengths and an angle measure, and encouraged to make triangles that do not look like one another’s. Given an angle and two sides that are not both adjacent to the given angle, there are three possible triangles that students can make.

As students negotiate how they will make the triangles, listen for different strategies students have for making different triangles. Monitor for students who:

• changed which side was adjacent to the given angle and which side was opposite
• swung the non-adjacent side around to see if there were multiple triangles that could be made
• found that only one triangle could be made when the longer side was the non-adjacent side

If a group of students decides it is only possible to make one or two different triangles, encourage them to list all the possible ways to order the given pieces and check that they have tried all of them. (Angle, short side, long side or angle, long side, short side)

Invite students to share how many different triangles their group was able to create. Select and sequence students to share their strategies who:

• changed which side was adjacent to the given angle and which side was opposite
• swung the non-adjacent side around to see if there were multiple triangles that could be made
• found that only one triangle could be made when the longer side was the non-adjacent side

In this activity, students are noticing and conjecturing. If no students observed when one triangle could be made and when two triangles could be made, that’s fine.

Ask students whether they think any two sides and one angle measure is enough information to guarantee that any copies of that triangle will be congruent. Ask them how their ideas are informed by the warm-up and this lesson.

• No, it’s not enough information because that’s what we knew about the warm-up triangle, and those triangles weren’t congruent.
• Sometimes it’s enough information, because the Side-Angle-Side Triangle Congruence Theorem works.
• Sometimes it’s enough information; when we had the longer side opposite the given angle, we could only make one triangle.

In the previous activity, students may have noticed that once they decided which side would be adjacent to the given angle, they could sometimes make two triangles and sometimes make one triangle. This may lead to questions about whether knowing an angle, a side, and another side (moving clockwise around the triangle) could be enough information to guarantee you can make a congruent copy of the triangle. This activity formalizes that question and answer by giving students instructions to make different triangles and exploring which instructions always lead to a single unique triangle and which are ambiguous.

 Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Identify a way for students to compare all the examples of a given triangle. For example, invite students to place all the triangles labeled \(ABC\) in a single visual display.

Arrange students in 8 groups. Provide each group with tools to create a visual display. Assign a different card to each group.

Display the prompt: “When you are given that two pairs of corresponding sides are congruent, and a pair of corresponding angles that are not between the sides are congruent, that is enough to guarantee triangle congruence if \(\underline{\hspace{.5in}}\), but not enough information if \(\underline{\hspace{.5in}}\).”

Invite students to do a gallery walk and determine how to fill in the blanks. (The longer side is opposite the angle...the shorter side is opposite the given angle.)

Note that triangles \(GHI\) and \(STU\) have the same side lengths and angle measures, just with a different ordering. Comparing these two cases might help students who are struggling to see the difference.

“Only one triangle can be made, and triangle congruence is guaranteed, when you know that the longer of the two given sides is opposite the given angle.” Add this theorem to the display of triangle congruence theorems.